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Introduction

1.1 The Piezoelectric Effect

In the most general terms, a material is piezoelectric if it transforms electrical into mechanical energy, and vice versa, in a reversible or lossless process. This transformation is evident at a macroscopic scale in what are commonly known as the direct and converse piezoelectric effects. The direct piezoelectric effect refers to the ability of a material to transform mechanical deformations into electrical charge. Equivalently, application of mechanical stress to a piezoelectric specimen induces flow of electricity in the direct piezoelectric effect. The converse piezoelectric effect describes the process by which the application of an electrical potential difference across a specimen results in its deformation. The converse effect can also be viewed as how the application of an external electric field induces mechanical stress in the specimen.

While the brothers Pierre and Jacques Curie discovered piezoelectricity in 1880, much the early impetus motivating its study can be attributed to the demands for submarine countermeasures that evolved during World War I. An excellent and concise history, before, during, and after World War I, can be found in [43]. With the increasing military interest in detecting submarines by their acoustic signatures during World War I, early research often studied naval applications, and specifically sonar. Paul Langevin and Walter Cady had pivotal roles during these early years. Langevin constructed ultrasonic transducers with quartz and steel composites. Shortly thereafter, the use of piezoelectric quartz oscillators became prevalent in ultrasound applications and broadcasting. The research by W.G. Cady was crucial in determining how to employ quartz resonators to stabilize high frequency electrical circuits.

A number of naturally occurring crystalline materials including Rochelle salt, quartz, topaz, tourmaline, and cane sugar exhibit piezoelectric effects. These materials were studied methodically in the early investigations of piezoelectricity. Following World War II, with its high demand for quartz plates, research and development of techniques to synthesize piezoelectric crystalline materials flourished. These efforts have resulted in a wide variety of synthetic piezoelectrics, and materials science research into specialized piezoelectrics continues to this day.

1.1.1 Ferroelectric Piezoelectrics

Perhaps one of the most important classes of piezoelectric materials that have become popular over the past few decades are the ferroelectric dielectrics. A ferroelectric can have coupling between the mechanical and electrical response that is several times a large as that in natural piezoelectrics. Ferroelectrics include materials such as barium titanate and lead zirconate titanate, and their unit cells are depicted in Figure 1.1. When the centers of positive and negative charge in a unit cell of a crystalline material do not coincide, the material is said to be polar or dielectric. An electric dipole moment is a vector that points from the center of negative charge to the center of positive charge, and its magnitude is equal to where is the magnitude of the charge at the centers and is the separation between the centers. The limiting volumetric density of dipole moments is the polarization vector . Intuitively we think of the polarization vector as measuring the asymmetry of the internal electric field of the piezoelectric crystal lattice. Ferroelectrics exhibit spontaneous electric polarization that can be reversed by the application of an external electric field. In other words, the polarization of the material is evident during a spontaneous process, one that evolves to a state that is thermodynamically more stable. Understanding this process requires a discussion of the micromechanics of a ferroelectric.

Figure 1.1 Barium titanate and lead zirconate titanate. (Left) Barium titanate with cation at the center, anions on the faces, and cations at the corners of the unit cell. (Right) Lead zircanate titanate with or cation at the center, anions on the faces, and cations at the corners of the unit cell.

The micromechanics of ferroelectric dielectrics is subtle and interesting. Above a critical temperature , the Curie temperature, the crystal structure of a ferroelectric is usually symmetric, and a plot of the polarization versus applied electric charge is generally nonlinear and single-valued as shown in Figure 1.2.

However, with cooling below the Curie temperature , a thermodynamic process drives a structural phase transition so that the final crystalline phase has a lower symmetry. At the lower temperature it can be shown [18] that the lower symmetry crystal phase has at least two energetically equivalent configurations or variants. Furthermore, with the application of an external electric field, it must be the case that it is possible switch among these crystalline variants in a reversible process. The ferroelectric material forms domains that consist of these energetically equivalent crystalline variants. Figure 1.3 depicts schematically the and domains [31] that can appear in single crystal barium titanate [31]. Note in the figure that the polarization vectors are opposite from one domain to the next, and their average polarization over a macroscale can have zero effective polarization. Because of the presence of these domains, below the Curie temperature the polarization versus applied electric field takes the form of a hysteresis loop as shown in Figure 1.4. Initially, the domains cancel their effects over the macroscopic specimen and at . The polarization increases as in Figure 1.2 for a range of electric field . When a critical value , the coercive electric field strength, is reached, the domains abruptly switch so that they are approximately well-aligned with the external electric field. With all domains having approximately aligned polarization vectors, the polarization again follows a nonlinear single valued curve until saturation is achieved. When the electric field is reversed, and reaches the opposite coercive electric field strength , the domains switch again so their polarization vectors are approximately aligned with the second variant. The result of this cyclic process is that after the transient response there is a nonzero polarization, the spontaneous polarization, for an electric field strength . At a macroscopic scale, then, the effective or average polarization can switch with the application of the external electric field.

Figure 1.2 Polarization versus applied electrical field for ferroelectric above the Curie temperature .

Figure 1.3 and domains in , [31].

Source: Walter J. Merz, Domain Formation and Domain Wall Motion in Ferro-electric BaTiO3 Single Crystals, em Physical Review, Volume 95, Number 3, August 1, 1954, pp. 690-698.

Figure 1.4 Polarization versus electrical field hysteresis below the Curie temperature .

1.1.2 One Dimensional Direct and Converse Piezoelectric Effect

In view of these observations, at a fundamental level, the micromechanics of piezoelectricity is understood in terms of crystalline asymmetry. While the most general theory of linear piezoelectricity of material continua in three dimensions is discussed in Chapter 5, intuition can be built by considering a one dimensional example. Figure 1.5 depicts the direct piezoelectric effect graphically, while the converse effect is shown in Figure 1.6. For the specimens shown, the mechanical variables are the stress and strain , and the electrical variables include the electric field , electric displacement , voltage , and the electrical potential . In Figure 1.5 we suppose that the top and bottom of the specimen are free to displace. A thin film electrode, one that does not alter the mechanical properties of the specimen, is applied to the top and bottom surfaces by a deposition or sputtering process. An ideal current meter, over which the potential difference is approximately zero, is attached to the top and bottom electroded surfaces. A positive stress is applied as shown. As we discuss in Chapter 5 the constitutive laws that couple the electrical and mechanical variables can take many forms. In this one dimensional example we choose to express the dependency among the electrical and mechanical variables as

where is the mechanical compliance constant, is the piezoelectric coupling constant, and is the permittivity constant. See Chapter 5 for a precise definition of these constants when interpreted as elements of tensors suitable for piezoelectric continua in three dimensions. Since the current meter is ideal, the electrical potential at the top electrode is equal to the potential at the bottom electrode, and the electric field . From Equation 1.1 it follows that the strain is positive, and the top and bottoms of the specimens displace by approximately , and the specimen stretches by a total amount . The magnitude of the charge on the electrodes can be obtained from the boundary conditions . Thus, we see that either the application of a positive stress , or equivalently of a positive strain , induces charges having magnitude on the faces of the specimen.

In contrast, Figure 1.6 illustrates how the application of an electric potential across a piezoelectric specimen generates a deformation. In this...